A lot of these equations I take from the review table.

And I can think of some, there's a list of looks that we're going to look at, one by

So they have all these abstract notations about the outgoing field is the free evolution without coupling to the lattice plus the zero point fluctuation of the test mass plus the signal and the response.

So that's where I learned all those things. So I have h bar cx, the combinator, tx of t prime, f of 0 of t prime plus the g of t prime.

So this is sort of what we did last time. And then this morning Ash gave a lecture where it's a different point of view.

If I rephrase it, what I have here is the high gain limit and therefore I have this species relation between the spectral density of z and f.

This is the first thing. And for me, because my gain is defined to be 1, that's why I have 1 here and this is the family here. And then the other thing is I ignored the zero point fluctuation of the test mass.

And therefore I derived a situation yesterday where there was no limits if you were able to do correlations between z and f.

In fact, in gravitational wave detectors in particular, because we're so far away from the zero point from the pendulum of motion of this mirror,

and because this mirror is a bar by cube, there's just no chance to reach the fundamental quantum limit of the other hand.

And there will be some kind of spin we will have either this or next lecture.

We'll see that an oscillator standard quantum limit or an oscillator quantum limit may also show up in the context of gravitational wave detection when the restoring force of that oscillator is not from the pendulum,

but actually from the radiation pressure force. So that we'll go into a little bit later.

But today, before I give you examples of what this is, let me review a little bit the quantum optics.

So I would like to establish some locations for my discussions. Therefore I will do quantum optics in a very quick way.

I'll basically say that I will have my electric field that is defined in this way in a one dimensional situation where I only consider a one dimensional situation.

And then I decompose this into moles, and then these moles I write down this kind of decomposition for my electric field.

It's a kind of a Gaussian unit system. So that was the forbidden conduit.

So this normalization is such that, okay so here the A is the cross section of my beam.

So I'm thinking about an electric field propagating along the X direction.

And then there's a certain cross sectional area for the beam and outside beam. There's nothing. And this is the mole I'm considering.

And the electric field is, the mole shape is like this. It's a prominent way, but it contains different, you can see from here this omega is a frequency.

But that's a very deceiving kind of a state. What you really have, okay this is a Heisberg omega.

What you really have for different omega are different spatial modes.

For each of these omega you either annihilate or create a spatial mode which would have a time limit oscillation frequency of omega.

But what it really has is a k which is equal to omega over c.

And sometimes I'm confused about c because I normally do c equal to 1.

So here, and also the normalization is such that the Hamiltonian is given by the volume integral of this e squared over 4 pi e to the X.

So here what about b? Well b and e are the same in this Gaussian system.

Okay, so this, and then after you write it down you'll find out that this Hamiltonian is actually equal to 0 to infinity h bar omega over 2 a dagger omega a omega plus a omega a omega dagger.

Which is consistent with what we have.

No, this is the photon number times e omega over 2 pi.

And then here we require the canonical computational relation that requires a omega and a omega prime dagger to be commuting to this kind of delta.

So this is h bar.

Yang Bai, you only keep the right movers here?

You only keep the right moving part of the...

Yes, yes. So here this is just one direction, one problem.

Yeah, this is the formation cardio. This will contain the creation of the...

Okay, so this, but for most of today and all the most of my lectures, we'll mostly consider fluctuations that are concentrated near one frequency, which is the carrier frequency.

So in the spectrum that we have here, we have positive frequency which corresponds to the annihilation operators.

And negative frequency, which are the creation operators. And then we mostly consider oscillations which are centered around omega 0 from a certain small region where the dynamics happens.

And then also minus omega 0 here.

Okay, so really the operators that we're concerned about are really a omega 0 plus omega and a dagger of omega 0 plus omega, where this omega, I guess we can say plus minus, and this omega is really taking place at a very small magnitude.

It's a dynamical time scale of your mechanical motion, but in my case it's very small.

Okay, so in this case sometimes it's convenient to reorganize these four things into four other things.

Okay, so you have omega 0 plus omega minus big omega, and you have these four things, four operators.

Then you can reorganize them into four other things, which are called the quadrature operators in the frequency domain. So you define a1 of omega as a of omega 0 plus omega plus a dagger of omega 0 minus omega square root 2.

And a2 is a omega 0 plus omega minus a dagger of omega 0 minus omega square root 2i.

Okay, so these two you define them, and then you can also, okay, so you have four of these guys.

Now you have this guy, this guy, this guy dagger, and this guy dagger. And here omega is, this guy dagger actually turns out to be this guy minus omega.